A Framework for Linear Prediction of Nonlinear Dynamical Systems Using Koopman Theory

Abstract

Despite many advances being made in classical techniques for handling dynamical systems,the class of nonlinear dynamical systems is yet to be treated under a ”one size fits all”scheme as is the case with linear dynamical systems. But given that all linear systems lendthemselves to easy representation, analysis and control, one could leverage existing theorythat allows us to examine nonlinear dynamical systems under the same lens.Koopman theory comes as an answer to this felt need of simplifying how we deal withnonlinear dynamical systems and influence their behaviour. By riding on the rising tidesof big data and massive compute that is prevalent in our times, data driven methods toapproximate the Koopman operator can be used to develop a framework to cast a nonlineardynamical system into a linear dynamical system in a higher dimensional state space. Thishigher dimensional state space can be operated in and the resulting actions and trajectoriesthat the system assumes in this higher state space can then be translated to the originalmanifold that the system lives in naturally.This thesis proposes an end-to-end framework that constructs a linear approximation toa nonlinear dynamical system by lifting the original state space to a higher dimensional state space where it is linear. The preprocessing stage that one must go through, the choiceof lifting function that results in the higher dimensional state space, building the linear modelin this higher dimensional state space and subsequently forecasting with an initial conditionand some control inputs, if applicable, are all discussed.Two methods were tried on three classes of systems which are unforced nonlinear dynamicalsystems, forced affine nonlinear dynamical systems and forced nonaffine nonlinear dynamicalsystems. One method leverage deep learning to choose the lifting functions to attain linearadvancement in the resulting higher dimensional state space and the other makes use ofsparse regression techniques to identify analytical expressions for possible candidate liftingfunctions, so in this case we are aware of what the lifting functions are exactly.The trajectories that were obtained from the linear model derived in the higher dimensionalstate space were very close to the original trajectories obtained by advancing the nonlinearsystem with a numerical solver. This shows that this single framework is very reliable inrepresenting and analyzing nonlinear dynamical systems.